\subsection{Fundamental Frequency}

The fundamental frequency is the lowest frequency which it is possible to generate using, for example, a stretched out string that is of a certain length. When finding the fundamental frequency you find the deepest tone which it is possible to generate with, for example, a certain instrument. Finding the fundamental frequency of something requires a few things. In situation where the user desires to find the fundamental frequency of a guitar string, all he or she is required is the length of the guitar string and it can be worked out. When calculating the fundamental frequency the following formula is used:

${f_{0}} = \frac{v}{\lambda}$

where ${f_{0}}$ is the fundamental frequency, $v$ is the velocity, and $\lambda$ is the wavelength.
Some of these variables is already known, for example when calculating the fundamental frequency of a string attached at both ends you know that the wavelength of the fundamental frequency will always be the length of the string times two ($2l$). This is because at the lowest frequency a string can have it will shift between being raised and lowered, like seen on figure \ref{fig:fundamental}.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.7\textwidth]{images/TheoryDesign/fundamental.png}
\caption{An illustration of a string vibrating at its lowest frequency - \cite{fundamental}}
\label{fig:fundamental}
\end{figure}

the wavelength of a wave is defined as the length between the start of a cycle and to the end, and when looking at figure \ref{fig:fundamental} you see half a wavecycle spanning over the length $l$, therefore the length of the wave must be $2l$.

As you are dealing with sound you also know that the velocity is the speed of sound. so a revised and more precise formula in this case will look like this:
${f_{0}} = \frac{363m/s}{2l}$

Example:
Given a guitar Sstring with a length of 60 cm calculate the fundamental frequency.

$v = 363m/s$

$\lambda = 2l = 2*60cm$


${f_{0}} = \frac{363m/s}{1.2m} = 302,5Hz$